Problem 54 - Algebra 1: Concepts and Skills

Question

Factor the expression completely. $$ c^{3}+2 c^{2}-8 c-16 $$

Step-by-Step Solution

Answer

The factorial expression of the given cubic polynomial is \((c + 2\sqrt{2})(c - 2\sqrt{2})(c + 2)\)

Step 1: Recognize the original expression

First off, look at the cubic polynomial in detail. The polynomial given is \(c^{3} + 2c^{2} - 8c - 16\).

Step 2: Group the terms

This cubic equation can be grouped into two distinct binomial expressions. Group the terms in such a way that common factors emerge in each binomial. The expression becomes \(c^{3} + 2c^{2}\) and \(-8c - 16\).

Step 3: Factor out common factors

Factor out the highest common factor from each binomial. This simplifies our expression to \(c^{2}(c + 2)\) and \(-8(c + 2)\).

Step 4: Apply the distributive law

Applying the distributive law means to rewrite the expression so that instead of lots of terms, we have a product of two factors. In this polynomials case expression become \((c^{2} - 8) * (c + 2)\).

Step 5: Factor the difference of squares

The (c^2 - 8) part of the expression is a difference of squares and can be factored further so that instead of \(c^{2} - 8\), we have \((c + 2\sqrt{2})(c - 2\sqrt{2})\). We have (c + 2\sqrt{2})(c - 2\sqrt{2})(c + 2) as the fully factored expression.

Key Concepts

Grouping Method in Algebra

When you come across a polynomial that doesn't factor at first glance, the grouping method can be your algebraic lifeline. It involves breaking down the polynomial into groups of terms that can each be factored separately. This becomes especially useful when dealing with cubic polynomials, like the one in our exercise, which appears daunting at first.
The key is to arrange the terms into pairs or groups where a common factor can be easily identified. Although the original exercise simply splits the cubic polynomial into two binomials, the grouping method can sometimes require rearranging the terms to find the optimum pairings. Always check if some rearrangement brings out common factors more prominently, making the subsequent steps smoother.
Once grouped, you'll find each pair of terms may share a common factor or perhaps even lend themselves to further factorization methods like the difference of squares.

Difference of Squares

The difference of squares is a simply beautiful pattern in algebra that pops up surprisingly often. It's when you have two terms, each a perfect square, separated by a subtraction sign. The general form looks like \(a^2 - b^2\).
The beauty lies in how it factors into \((a + b)(a - b)\). The exercise above showcases this perfectly with \(c^2 - 8\). Notice how 8 is a square number because it's \(2\root{2}\times2\root{2}\), making it possible to neatly split our terms into \((c + 2\root{2})(c - 2\root{2})\). Recognizing this pattern is a fundamental skill in algebra as it simplifies what could be complicated problems into something far more manageable.

Distributive Law in Algebra

The distributive law, sometimes known as the distributive property, is an algebraic lifeline that allows us to multiply through parentheses elegantly. It serves to simplify expressions and to expand polynomials.
In the context of our cubic polynomial, once we've pulled out the common factor, what remains are binomials that we can't just add together. By applying the distributive law, we arrive at a product of two terms, which in this case are \((c^2 - 8)\) and \((c + 2)\). It's essential to remember that the distributive property works both ways, so if you ever need to expand or factor expressions, this method will be your guide.

Common Factor Extraction

When you're face-to-face with a complex polynomial, one of your first algebraic instincts should be to sniff out any common factors. Extracting common factors is like finding the golden thread that strings together various parts of the polynomial.
In our exercise, we identified \(c^2\) as a common factor in the first binomial and \(-8\) in the second. Pulling them out simplifies the expression significantly. This process reduces the polynomial and unveils a simpler structure hidden within. It's a crucial tool to wield in your journey through algebra and makes deciphering the true nature of polynomials that much easier. Always look for the highest common factor, as it will provide the cleanest and most efficient simplification.